NONMONOTONIC FUZZY MEASURES AND THE CHOQUET INTEGRAL

This paper discusses non-monotonic fuzzy measures, which are set funct ions without monotonicity, and the Choquet integral with respect to no n-monotonic fuzzy measures. A concrete example shows that the Choquet integral is meaningful in the case of non-monotonic fuzzy measures as well as in the case of ordinary fuzzy measures. Every comonotonically additive, positively homogeneous functional of bounded variation can b e represented as a Choquet integral with respect to a non-monotonic fu zzy measure of bounded variation. The space of such functionals is a r eal Banach space isometrically isomorphic to the space of non-monotoni c fuzzy measures of bounded variation.